APPROACHES TO FUZZY
LOGIC MINIMIZATION
Graphical Representations
• Kandel's and Francioni's Approach
• Fuzzy to Multiple-valued Function Conversion
Approach
• Fuzzy Logic Decision Diagrams Approach
• Fuzzy Logic Multiplexer
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Approaches to Fuzzy Logic Minimization: Graphical Representations and Decomposition, Slides of Robotics
Various approaches to fuzzy logic minimization, including kandel's and francioni's approach, fuzzy to multiple-valued function conversion, fuzzy logic decision diagrams approach, and fuzzy logic multiplexer. The text also covers graphical representations such as fuzzy maps, lattice of two variables, and the subsumption rule. It provides examples and formulas for decomposing fuzzy functions using sum-of-products and canonical figures.
Typology: Slides
2012/2013
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APPROACHES TO FUZZY
LOGIC MINIMIZATION
Graphical Representations
- Fuzzy to Multiple-valued Function Conversion• Kandel's and Francioni's Approach
Approach
- Fuzzy Logic Multiplexer• Fuzzy Logic Decision Diagrams Approach
Graphical Representations
- Form to Decompose a Fuzzy• The Subsumption rule• Lattice of two variables• Fuzzy Maps
Functions
Lattice of Two Variables
terms.possibleof all therelationshipShows the
term.to a singlebe reducedterms canwhich twoShows
rule is:subsumptionThe function.a fuzzy logicUsed to reduce α
xi
x’
I
β
α
x
i
x’
I
β
= x
i
x’
I
subsuming i.with Imap are showntwo variable Operations on β
The
The Subsumption
Subsumption Rule
Rule
Form Needed to Decompose
Fuzzy Functions
- Form requirements:
Sum-of-products
x2 Figures show the functionCanonical
x’
2
x’
1
x
x
x’
2
x
x’
1
x’
2
before using the subsuming
x’ rules in (a) and after in (d)
1 x 2 + x 1
x’
x
x
x’
x’
x
xx
x’x’
xx
x’x’
x’x’
=
x
x’
x’
x
xx
x’x’
(1+
x’
)
= x
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x’
x
xx
x’x’
= x’
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x’
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2 2.^
αααααααα
xx
ii
x
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’
II^
βββββ
βββ
+
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’
’ x
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x
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’
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=
= x
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ββββββββ
Kandel's and Francioni's
Approach
- Example using Kandel and Francioni• S-Maps• Variable Matching DIP’s Table• Decomposition Implicant Pattern (DIP)
approach Example
- Fuzzy Logic Circuits from Example– Second Decomposition in Example
Decomposition Implicant Pattern
(DIP)
S-Maps
Arrange two-variable fuzzy maps for n variables.
S-map.This method is just done by iteration to form an n variable
This shows X
1
is made up of repeated X
2
and X
3
two
variable maps.
Example using Kandel and
Francioni approach
f = F[(G(w,y), x,z)]and G’(w,y) = G’(Y)By substituting: G(w,y) = G(Y)f = (wy)z + (w’ + y’) x’zz’ + xzg(w,y) =wy, G’ (w,y) = w’ + y’From DIP 1 implies: f = x’y’zz’ + xz + w’x’zz’ + wyz
= G(Y)z + G’(Y) x’zz’ + xz
Fuzzy Logic Circuits from
Example
decomposition.(a) First level of
decomposition.(b) Second level of
decomposition.second level of(c) Result of first and
(d) Original function.
APPROACHES TO FUZZY LOGIC DECOMPOSITION
• Kandel's and Francioni's Approach• Graphical Representations
Conversion ApproachFuzzy to Multiple-valued Function
- Fuzzy Logic Multiplexer• Fuzzy Logic Decision Diagrams Approach
Fuzzy Function Ternary Map
This shows the mapping between the fuzzy terms and terms in the ternary map.
Fuzzy Function to Three-valued Function
ConversionConversion
Example
function terms:the Fuzzy Conversion of
x
x’
2
x’
1
x
x
x’
2
x
x’
1
x
f = x2shown fromoperation asusing the MINcanonical formIn non-
x’
2
+x’
1
x
+x
1
x’
2
x
x’
1
x’
2